Mode Coupling between the Nonpolar K$_{3}$ and Polar $\Gamma _{2}^{-}$ Phonons as the Ferroelectricity Origin of Multiferroic h-LuMnO$_{3}$

LuMnO$_{3}$ is expected to show the highest stability towards the hexagonal phase among 15 different lanthanide-based manganites. Currently, the most puzzling problem associated with the hexagonal LuMnO$_{3}$ (h-LMO) is the observed large temperature-gap between the structural phase transition to the polar P6$_{3}$cm phase at $\sim$ 1290 K and the emergence of the spontaneous polarization at a substantially reduced temperature, $\sim$ 750 K. This anomalous temperature-gap has also been observed in h-YMnO$_{3}$. To resolve this puzzling issue, we have carried out density-functional theory calculations and found that the structural phase transition to the polar P6$_{3}$ cm phase from the nonpolar P63/mmc phase is mediated by the freezing-in of the zone-boundary K$_{3}$ phonon in h-LMO. However, the spontaneous ferroelectric polarization does not appear until the amplitude of K$_{3}$ phonon becomes a certain critical value above which the coupling of the polar $\Gamma_{2}^{-}$ mode with the nonpolar K$_{3}$ mode is practically turned on. This mode-coupling-induced polarization, thus, elucidates the above puzzle.